Monge-Ampere equations in big cohomology classes

We define non-pluripolar products of an arbitrary number of closed positive (1, 1)-currents on a compact Kahler manifold X. Given a big (1, 1)-cohomology class alpha on X (i.e. a class that can be represented by a strictly positive current) and a positive measure mu on X of total mass equal to the volume of alpha and putting no mass on pluripolar sets, we show that mu can be written in a unique way as the top-degree self-intersection in the non-pluripolar sense of a closed positive current in alpha. We then extend Kolodziedj's approach to sup-norm estimates to show that the solution has minimal singularities in the sense of Demailly if mu has L (1+epsilon) -density with respect to Lebesgue measure. If mu is smooth and positive everywhere, we prove that T is smooth on the ample locus of alpha provided alpha is nef. Using a fixed point theorem, we finally explain how to construct singular Kahler-Einstein volume forms with minimal singularities on varieties of general type.